fill in the missing digits in as many ways as
possible so the number will be divisible by 6.
3_2_
A number is divisible by 6 if it is divisible by 2 AND by 3
Clearly since the above number is even, it is divisible by 2, so all we have to worry about is the 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
So far we have a sum of 5.
So adding 1 or 4 or 7 would leave the sum divisible by 3
so the missing digits could be 1, 4, or 7
(numbers are 312, 342, or 372, check to see if they divide by 6)
To find the missing digits in the number 3_2_, we need to determine the possible values that can be inserted to make the number divisible by 6.
To be divisible by 6, a number needs to be divisible by both 2 and 3.
- Divisibility by 2: To be divisible by 2, the last digit of the number needs to be even. Possible even digits are 0, 2, 4, 6, or 8.
- Divisibility by 3: The sum of all the digits in the number needs to be divisible by 3. To simplify the process, we can find a pattern by examining the possible two-digit numbers formed by inserting digits in the missing positions.
- If we insert 0 in the first missing position, we have 302_. The sum is 3 + 0 + 2 + _ = 5 + _.
- If we insert 2 in the first missing position, we have 322_. The sum is 3 + 2 + 2 + _ = 7 + _.
- If we insert 4 in the first missing position, we have 342_. The sum is 3 + 4 + 2 + _ = 9 + _.
- If we insert 6 in the first missing position, we have 362_. The sum is 3 + 6 + 2 + _ = 11 + _.
- If we insert 8 in the first missing position, we have 382_. The sum is 3 + 8 + 2 + _ = 13 + _.
Now, we can examine the sums and look for patterns to determine the possible values. Since we want a sum that is divisible by 3, the second digit (_ in each case) needs to be a number from 0 to 9 that makes the sum divisible by 3.
- For a sum of 5 + _, any number from 1 to 4 would make the sum divisible by 3. So, for 302_, the possible values for the second digit are 1, 2, 3, or 4.
- For a sum of 7 + _, any number from 2 to 1 would make the sum divisible by 3. So, for 322_, the possible values for the second digit are 2, 5, or 8.
- For a sum of 9 + _, any number from 0 to 9 would make the sum divisible by 3. So, for 342_, any number from 0 to 9 can be in the second digit.
- For a sum of 11 + _, any even number would make the sum divisible by 3. So, for 362_, the possible values for the second digit are 0, 2, 4, 6, or 8.
- For a sum of 13 + _, any number from 1 to 2 would make the sum divisible by 3. So, for 382_, the possible values for the second digit are 1.
Hence, the possible numbers that are divisible by 6 are:
3020, 3022, 3024, 3026, 3028
3225, 3228
3420, 3421, 3422, 3423, 3424, 3425, 3426, 3427, 3428, 3429
3620, 3622, 3624, 3626, 3628
3821